Suppose that we are given \(A(0)\in {\mbox{$\mathbb {R}$}}^{m\times m}\)  with \(A_{ij}\ge 0\)  for all \( i,j\in [m]\)  and \(\Vert A(0)\Vert \le \lambda _2\)  for some positive constant \(\lambda _2\) . By choosing \(\gamma >0\)  small enough for which \(\gamma _0=\gamma \lambda _2 < 1\) , Lemma REF  implies that \(\alpha _0:=\sigma _{\min }[Z(0)]\ge \sigma _{\min }[\Phi (0)]/(1+\gamma _0)\) . Therefore, the satisfiability of the assumption made in Theorem REF  is reduced to find an appropriate \(\Phi (0)\)  or \(W(0)\)  for which \(\sigma _{\min }[\Phi (0)]/(1+\gamma _0)\)  satisfies the initial conditions (REF )-(). Some mostly recent literature [1]}, [2]}, [3]}, [4]}, [5]} have been provided rigorously theoretical analyses and concrete examples to demonstrate that the initial conditions are easily satisfied with different types of initialization. The rest of this section provides concrete examples to show a subset of initialization that satisfies these assumptions.
